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The formalism of density operators and matrices was introduced in 1927 by John von Neumann [29] and independently, but less systematically, by Lev Landau [30] and later in 1946 by Felix Bloch. [31] Von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.
For Liouville's equation in quantum mechanics, see Von Neumann equation. For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation . In differential geometry , Liouville's equation , named after Joseph Liouville , [ 1 ] [ 2 ] is the nonlinear partial differential equation satisfied by the conformal factor f of a ...
The Schrödinger equation or, actually, the von Neumann equation, is a special case of the GKSL equation, which has led to some speculation that quantum mechanics may be productively extended and expanded through further application and analysis of the Lindblad equation. [2]
The quantum Liouville equation is the Weyl–Wigner transform of the von Neumann evolution equation for the density matrix in the Schrödinger representation. The quantum Hamilton equations are the Weyl–Wigner transforms of the evolution equations for operators of the canonical coordinates and momenta in the Heisenberg representation.
The analog of Liouville equation in quantum mechanics describes the time evolution of a mixed state. Canonical quantization yields a quantum-mechanical version of this theorem, the von Neumann equation. This procedure, often used to devise quantum analogues of classical systems, involves describing a classical system using Hamiltonian mechanics.
In mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system.
The Liouville–Neumann series is defined as ϕ ( x ) = ∑ n = 0 ∞ λ n ϕ n ( x ) {\displaystyle \phi \left(x\right)=\sum _{n=0}^{\infty }\lambda ^{n}\phi _{n}\left(x\right)} which, provided that λ {\displaystyle \lambda } is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of ...
The phase-space formulation is a formulation of quantum mechanics that places the position and momentum variables on equal footing in phase space.The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product.